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Theorem frgrwopreglem5lem 28197
 Description: Lemma for frgrwopreglem5 28198. (Contributed by AV, 5-Feb-2022.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
frgrwopreg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgrwopreglem5lem (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → ((𝐷𝑎) = (𝐷𝑥) ∧ (𝐷𝑎) ≠ (𝐷𝑏) ∧ (𝐷𝑥) ≠ (𝐷𝑦)))
Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝐴,𝑏   𝑥,𝐵   𝑦,𝐷   𝐺,𝑎,𝑏,𝑦,𝑥   𝑦,𝑉
Allowed substitution hints:   𝐴(𝑦,𝑎)   𝐵(𝑦,𝑎,𝑏)   𝐷(𝑎,𝑏)   𝐸(𝑥,𝑦,𝑎,𝑏)   𝐾(𝑦,𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem frgrwopreglem5lem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 frgrwopreg.a . . . . . 6 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
21rabeq2i 3401 . . . . 5 (𝑥𝐴 ↔ (𝑥𝑉 ∧ (𝐷𝑥) = 𝐾))
3 fveqeq2 6668 . . . . . . 7 (𝑥 = 𝑎 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑎) = 𝐾))
43, 1elrab2 3606 . . . . . 6 (𝑎𝐴 ↔ (𝑎𝑉 ∧ (𝐷𝑎) = 𝐾))
5 eqtr3 2781 . . . . . . . . 9 (((𝐷𝑎) = 𝐾 ∧ (𝐷𝑥) = 𝐾) → (𝐷𝑎) = (𝐷𝑥))
65expcom 418 . . . . . . . 8 ((𝐷𝑥) = 𝐾 → ((𝐷𝑎) = 𝐾 → (𝐷𝑎) = (𝐷𝑥)))
76adantl 486 . . . . . . 7 ((𝑥𝑉 ∧ (𝐷𝑥) = 𝐾) → ((𝐷𝑎) = 𝐾 → (𝐷𝑎) = (𝐷𝑥)))
87com12 32 . . . . . 6 ((𝐷𝑎) = 𝐾 → ((𝑥𝑉 ∧ (𝐷𝑥) = 𝐾) → (𝐷𝑎) = (𝐷𝑥)))
94, 8simplbiim 509 . . . . 5 (𝑎𝐴 → ((𝑥𝑉 ∧ (𝐷𝑥) = 𝐾) → (𝐷𝑎) = (𝐷𝑥)))
102, 9syl5bi 245 . . . 4 (𝑎𝐴 → (𝑥𝐴 → (𝐷𝑎) = (𝐷𝑥)))
1110imp 411 . . 3 ((𝑎𝐴𝑥𝐴) → (𝐷𝑎) = (𝐷𝑥))
1211adantr 485 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑎) = (𝐷𝑥))
13 frgrwopreg.v . . . 4 𝑉 = (Vtx‘𝐺)
14 frgrwopreg.d . . . 4 𝐷 = (VtxDeg‘𝐺)
15 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
1613, 14, 1, 15frgrwopreglem3 28191 . . 3 ((𝑎𝐴𝑏𝐵) → (𝐷𝑎) ≠ (𝐷𝑏))
1716ad2ant2r 747 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑎) ≠ (𝐷𝑏))
18 fveqeq2 6668 . . . . . 6 (𝑥 = 𝑧 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑧) = 𝐾))
1918cbvrabv 3405 . . . . 5 {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = {𝑧𝑉 ∣ (𝐷𝑧) = 𝐾}
201, 19eqtri 2782 . . . 4 𝐴 = {𝑧𝑉 ∣ (𝐷𝑧) = 𝐾}
2113, 14, 20, 15frgrwopreglem3 28191 . . 3 ((𝑥𝐴𝑦𝐵) → (𝐷𝑥) ≠ (𝐷𝑦))
2221ad2ant2l 746 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑥) ≠ (𝐷𝑦))
2312, 17, 223jca 1126 1 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → ((𝐷𝑎) = (𝐷𝑥) ∧ (𝐷𝑎) ≠ (𝐷𝑏) ∧ (𝐷𝑥) ≠ (𝐷𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  {crab 3075   ∖ cdif 3856  ‘cfv 6336  Vtxcvtx 26881  Edgcedg 26932  VtxDegcvtxdg 27347 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-iota 6295  df-fv 6344 This theorem is referenced by:  frgrwopreglem5  28198
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